A screening-based gradient-enhanced Kriging modeling method for high-dimensional problems

By exploring the auxiliary information from gradients, the accuracy of Kriging model can be improved. However, the dramatically increased time for model training tends to be unaffordable. Therefore, a novel gradient-enhanced Kriging modeling method which utilizes only a partial set of gradients, is developed in this article. Within the framework of this method, a balance between model accuracy and modeling efficiency can be achieved. More specifically, the influence of each input variable on output is estimated and ranked by feature selection technique. Then an empirical evaluation rule is proposed to facilitate the selection of gradients. Five representative numerical benchmarks from 10-D to 30-D and an airfoil optimal shape design with 18 variables are used for validation. Results show that when compared with the conventional Gradient-enhanced Kriging, the modeling time of the proposed method is significantly reduced, while the loss of accuracy is negligible. As a consequence, the proposed surrogate modeling method can provide an alternative way for approximating high-dimensional problems.     

Modified screening-based Kriging method with cross validation and application to engineering design

In this paper, a basis screening Kriging method using cross validation error is proposed to alleviate computational burden of the dynamic Kriging while maintaining its accuracy. Metamodeling is widely used for design optimization of complex engineering applications where considerable computation time is required. The Kriging method is one of popular metamodeling methods due to its accuracy and efficiency. There have been many attempts to improve accuracy of the Kriging method, and the dynamic Kriging method using cross-validation error, which selects adequate basis functions to best describe the mean structure of a response using a genetic algorithm, achieves outstanding performance in terms of accuracy. However, despite its accuracy, the dynamic Kriging requires very large amounts of computation because of the genetic algorithm and no limitation for order of basis functions. In the proposed method, a basis function set is determined by screening each basis function instead of using the genetic algorithm, which has advantages in computation for high dimensional metamodels or repeated metamodel generation. Numerical studies with four mathematical examples and two engineering applications verify that the proposed basis screening Kriging significantly reduces computation time with similar accuracy as the dynamic Kriging.     

A new method for reliability analysis of structures with mixed random and convex variables

This paper proposes a method combining projection-outline-based active learning strategy with Kriging metamodel for reliability analysis of structures with mixed random and convex variables. In this method, it is determined that the approximation accuracy of projection outlines on the limit-state surface is crucial for estimation of failure probability instead of the whole limit-state surface. To efficiently improve the approximation accuracy of projection outlines, a new projection-outline-based active learning strategy is developed to sequentially obtain update points located around the projection outlines. Taking into account the influence of metamodel uncertainty on the estimation of failure probability, a quantification function of metamodel uncertainty is developed and introduced in the stopping condition of Kriging metamodel update. Finally, Monte Carlo simulation is employed to calculate the failure probability based on the refined Kriging metamodel. Four examples including the Burro Creek Bridge and a piezoelectric energy harvester are tested to validate the performance of the proposed method. Results indicate that the proposed method is accurate and efficient for reliability analysis of structures with mixed random and convex variables.     

Non-Probabilistic Structural Reliability Analysis Integrating the PSO and the Kriging Model北大核心CSCD

针对复杂结构可靠性分析中面临的隐式功能函数和小样本问题,提出了一种粒子群优化和Kriging模型相结合的结构非概率可靠性分析方法。采用多维椭球描述结构不确定参数,运用粒子群优化对模型相关参数进行求解,并构建隐式功能函数的Kriging模型进行可靠性分析。三个算例结果表明所提方法有效可行,精度和效率均优于基于Kriging模型的非概率可靠性分析方法。     

Semiparametric inference for transformation models via empirical likelihood

Recent advances in the transformation model have made it possible to use this model for analyzing a variety of censored survival data. For inference on the regression parameters, there are semiparametric procedures based on the normal approximation. However, the accuracy of such procedures can be quite low when the censoring rate is heavy. In this paper, we apply an empirical likelihood ratio method and derive its limiting distribution via U-statistics. We obtain confidence regions for the regression parameters and compare the proposed method with the normal approximation based method in terms of coverage probability. The simulation results demonstrate that the proposed empirical likelihood method overcomes the under-coverage problem substantially and outperforms the normal approximation based method. The proposed method is illustrated with a real data example. Finally, our method can be applied to general U-statistic type estimating equations.     

Gradient-based approximation methods applied to the optimal design of vehicle suspension systems using computational models with severe inherent noise

The principal aim of this paper is to evaluate the feasibility of using gradient-based approximation methods for the optimisation of the spring and damper characteristics of an off-road vehicle, for both ride comfort and handling. The Sequential Quadratic Programming algorithm and the relatively new Dynamic-Q method are the two successive approximation methods used for the optimisation. The determination of the objective function value is performed using computationally expensive numerical simulations that exhibit severe inherent numerical noise. The use of forward finite differences and central finite differences for the determination of the gradients of the objective function within Dynamic-Q is also investigated. This is done in investigating methods for overcoming the difficulties associated with the optimisation of noisy objective functions.A recreational off-road vehicle is modelled in ADAMS, and coupled to MATLAB for the execution of the optimisation process. The full vehicle ADAMS model includes suspension kinematics, a load-dependent tyre model, as well as non-linear springs and dampers. Up to four design variables are considered in modelling the suspension characteristics.It is found that both algorithms perform well in optimising handling. However, difficulties are encountered in obtaining improvements in the design process when ride comfort is considered. Nevertheless, meaningful design configurations are still achievable through the proposed optimisation process, at a relatively low cost in terms of the number of simulations that have to be performed.     

Diseño óptimo robusto utilizando modelos Kriging: aplicación al diseño óptimo robusto de estructuras articuladas

Conventional methods addressing the robust design optimization problem of structures usually require high computational requirements due to the nesting of uncertainty quantification within the optimization process. In order to address such a problem, this work proposes a methodology, based on Kriging models, to efficiently assess the uncertainty quantification in the optimization process. The Kriging model approximates the structural performance both in the design domain and in the stochastic domain, which allows to decouple the uncertainty quantification process and the optimization process. In addition, an infill criterion based on the variance of the Kriging prediction is included to update the Kriging model towards the global Pareto front. Three numerical examples show the applicability and the accuracy of the proposed methodology. The results show that the proposed method is appropriate to solve the robust design optimization problem with reasonable accuracy and a considerably lower number of function calls than required by conventional methods.     

An efficient multi-objective PSO algorithm assisted by Kriging metamodel for expensive black-box problems

The huge computational overhead is the main challenge in the application of community based optimization methods, such as multi-objective particle swarm optimization and multi-objective genetic algorithm, to deal with the multi-objective optimization involving costly simulations. This paper proposes a Kriging metamodel assisted multi-objective particle swarm optimization method to solve this kind of expensively black-box multi-objective optimization problems. On the basis of crowding distance based multi-objective particle swarm optimization algorithm, the new proposed method constructs Kriging metamodel for each expensive objective function adaptively, and then the non-dominated solutions of the metamodels are utilized to guide the update of particle population. To reduce the computational cost, the generalized expected improvements of each particle predicted by metamodels are presented to determine which particles need to perform actual function evaluations. The suggested method is tested on 12 benchmark functions and compared with the original crowding distance based multi-objective particle swarm optimization algorithm and non-dominated sorting genetic algorithm-II algorithm. The test results show that the application of Kriging metamodel improves the search ability and reduces the number of evaluations. Additionally, the new proposed method is applied to the optimal design of a cycloid gear pump and achieves desirable results.     

A novel single-loop procedure for time-variant reliability analysis based on Kriging model

This paper proposes a novel single-loop procedure for time-variant reliability analysis based on a Kriging model. A new strategy is presented to decouple the double-loop Kriging model for time-variant reliability analysis, in which the extreme value response in double-loop procedure is replaced by the best value in the current sampled points to avoid the inner optimization loop. Consequently, the extreme value response surface for time-variant reliability analysis can be directly established through a single-loop Kriging surrogate model. To further improve the accuracy of the proposed Kriging model, two methods are provided to adaptively choose a new sample point for updating the model. One method is to apply two commonly used learning functions to select the new sample point that resides as close to the extreme value response surface as possible, and the other is to apply a new learning function to select the new point. Synchronously, the corresponding different stopping criteria are also provided. It is worth nothing that the proposed single-loop Kriging model for time-variant reliability analysis is for a single time-variant performance function. To verify the proposed method, it is applied to four examples, two of which have with random process and others have not. Other popular methods for time-variant reliability analysis including the existing single-loop Kriging model are also used for the comparative analysis and their results testify the effectiveness of the proposed method.     

Free vibration and buckling analysis of plates using B-spline wavelet on the interval Mindlin element

A finite element method (FEM) of B-spline wavelet on the interval (BSWI) is used in this paper to solve the free vibration and buckling problems of plates based on Reissner–Mindlin theory. By aid of the high accuracy of B-spline functions approximation for structural analysis, the proposed method could obtain a fast convergence and a satisfying numerical accuracy with fewer degrees of freedoms (DOF). The numerical examples demonstrate that the present BSWI method achieves the high accuracy compared to the exact solution and others existing approaches in the literatures. The BSWI finite element has potential to be used as a numerical method in analysis and design.     

An Inverse Reliability-based Approach for Designing under Uncertainty with Application to Robust Piston Design

In this work, we propose an optimization framework for designing under uncertainty that considers both robustness and reliability issues. This approach is generic enough to be applicable to engineering design problems involving nonconvex objective and constraint functions defined in terms of random variables that follow any distribution. The problem formulation employs an Inverse Reliability Strategy that uses percentile performance to address both robustness objectives and reliability constraints. Robustness is achieved through a design objective that evaluates performance variation as a percentile difference between the right and left trails of the specified goals. Reliability requirements are formulated as Inverse Reliability constraints that are based on equivalent percentile performance levels. The general proposed approach first approximates the formulated problem via a Gaussian Kriging model. This is then used to evaluate the percentile performance characteristics of the different measures inherent in the problem formulation for various design variable settings via a Most Probable Point of Inverse Reliability search algorithm. By using these percentile evaluations in concert with the response surface methodology, a polynomial programming approximation is generated. The resulting problem formulation is finally solved to global optimality using the Reformulation–Linearization Technique (RLT) approach. We demonstrate this overall proposed approach by applying it to solve the problem of reducing piston slap, an undesirable engine noise due to the secondary motion of a piston within a cylinder.     

An Improved Hybrid Adjoint Method in External Aerodynamics Using Variational Technique for the Boundary Integral Based Optimal Objective Function Gradient

An improved hybrid adjoint method to the viscous, compressible Reynold-Averaged Navier-Stokes Equation (RANS) is developed for the computation of objective function gradient and demonstrated for external aerodynamic design optimization. In this paper, the main idea is to extend the previous coupling of the discrete and continuous adjoint method by the grid-node coordinates variation technique for the computation of the variation in the gradients of flow variables. This approach in combination with the Jacobian matrices of flow fluxes refrained the objective function from field integrals and coordinate transformation matrix. Thus, it opens up the possibility of employing the hybrid adjoint method to evaluate the subsequent objective function gradient analogous to many shape parameters, comprises of only boundary integrals. This avoids the grid regeneration in the geometry for every surface perturbation in a structured and unstructured grid. Hence, this viable technique reduces the overall CPU cost. Moreover, the new hybrid adjoint method has been successfully applied to the computation of accurate sensitivity derivatives. Finally, for the investigation of the presented numerical method, simulations are carried out on NACA0012 airfoil in a transonic regime and its accuracy and effectiveness related to the new gradient equation have been verified with the Finite Difference Method (FDM). The analysis reveals that the presented methodology for the optimization provides the designer with an indispensable CPU-cost effective tool to reshape the complex geometry airfoil surfaces, useful relative to the state-of-the-art, in a less computing time.     

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.     

A computational framework of kinematic accuracy reliability analysis for industrial robots

A new computational method to evaluate comprehensively the positional accuracy reliability for single coordinate, single point, multipoint and trajectory accuracy of industrial robots is proposed using the sparse grid numerical integration method and the saddlepoint approximation method. A kinematic error model of end-effector is constructed in three coordinate directions using the sparse grid numerical integration method considering uncertain parameters. The first-four order moments and the covariance matrix for three coordinates of the end-effector are calculated by extended Gauss–Hermite integration nodes and corresponding weights. The eigen-decomposition is conducted to transform the interdependent coordinates into independent standard normal variables. An equivalent extreme value distribution of response is applied to assess the reliability of kinematic accuracy. The probability density function and probability of failure for extreme value distribution are then derived through the saddlepoint approximation method. Four examples are given to demonstrate the effectiveness of the proposed method.     

基于Kriging模型和提升小波变换的随机模型修正北大核心CSCD

为提高随机模型修正效率,减小计算量,提出了一种基于Kriging模型和提升小波变换的随机模型修正方法.首先,对加速度频响函数进行提升小波变换,提取第5层近似系数代替原频响函数.其次,采用拉丁超立方抽样抽取待修正样本,将其作为Kriging模型的输入,对应的近似系数作为输出,构建Kriging模型.提出了一种引入莱维飞行(Lévy flight)的蝴蝶优化算法(LBOA),并将其应用于Kriging模型相关参数的寻优中,提高Kriging模型的精度.最后,以最小化Wasserstein距离为目标,通过鲸鱼优化算法求解待修正参数的均值.测试函数结果表明,LBOA在寻优能力、收敛精度和稳定性等方面有了很大的提升.数值算例的修正误差均低于0.4%,验证了所提模型修正方法具有较高的修正精度和效率.     

A Derivative-Free Algorithm for Bound Constrained Optimization

In this work, we propose a new globally convergent derivative-free algorithm for the minimization of a continuously differentiable function in the case that some of (or all) the variables are bounded. This algorithm investigates the local behaviour of the objective function on the feasible set by sampling it along the coordinate directions. Whenever a suitable descent feasible coordinate direction is detected a new point is produced by performing a linesearch along this direction. The information progressively obtained during the iterates of the algorithm can be used to build an approximation model of the objective function. The minimum of such a model is accepted if it produces an improvement of the objective function value. We also derive a bound for the limit accuracy of the algorithm in the minimization of noisy functions. Finally, we report the results of a preliminary numerical experience.     

单调全局最优化问题的凸化外逼近算法

单调优化是指目标函数与约束函数均为单调函数的全局优化问题．本文提出一种新的凸化变换方法把单调函数化为凸函数，进而把单调优化问题化为等价的凸极大或凹极小问题，然后采用Hoffman的外逼近方法来求得问题的全局最优解．我们把这种凸化方法同Tuy的Polyblock外逼近方法作了比较，通过数值比较可以看出本文提出的凸化的方法在收敛速度上明显优于Polyblock方法．     

应力和位移约束下连续体结构拓扑优化

同时考滤应力和位移约束的连续体结构拓扑优化问题,很难用现有的均匀方法或变密度方法等求解。主要困难在于难以建立应力和位移约束与拓扑设计变量间显式关系式;即使建立了这种关系,也由于优化问题规模过大,利用常规的数学规划方法难以求解。隋允康、杨德庆曾提出了基于独立连续拓扑变量及映射变换（ＩＣＭ）的桁架结构拓扑优化模型。本文在此基础上,建立了以重量为目标,考虑应力和位移约束的连续体结构拓扑优化模型,并推导出     

Local topological equivalence of nonsmooth functions in Hilbert spaces

In this paper we investigate the relation between nonsmooth functions with domain in a Hilbert space and their local approximations. We consider Lipschitz functions and define an approximation model with directional derivatives. The qualitative behaviour of the approximation is studied by means of the concept of topological equivalence. Using this concept we establish the existence of a local coordinate transformation between the original function and the local approximation.     

Adaptive surrogate-based design optimization with expected improvement used as infill criterion

A prominent advantage of using surrogate models in structural design optimization is that computational effort can be greatly reduced without significantly compromising model accuracy. The essential goal is to perform the design optimization with fewer evaluations of the typically finite element analysis and ensuring accuracy of the optimization results. An adaptive surrogate based design optimization framework is proposed, in which Latin hypercube sampling and Kriging are used to build surrogate models. Accuracy of the models is improved adaptively using an infill criterion called expected improvement (EI). It is the anticipated improvement that an interpolation point will lead to the current surrogate models. The point that will lead to the maximum EI is searched and used as infill points at each iteration. For constrained optimization problems, the surrogate of constraint is also utilized to form a constrained EI as the corresponding infill criterion. Computational trials on mathematical test functions and on a three-dimensional aircraft wing model are carried out to test the feasibility of this method. Compared with the traditional surrogate base design optimization and direct optimization methods, this method can find the optimum design with fewer evaluations of the original system model and maintain good accuracy.